A316655 - OEIS

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A316655

Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

A rooted tree is series-reduced if every non-leaf node has at least two branches.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

LINKS

Table of n, a(n) for n=1..40.

FORMULA

a(prime(n)) = A000669(n).

a(2^n) = A000311(n).

EXAMPLE

Sequence of sets of trees begins:

2: 1

3: (11)

4: (12)

5: (1(11)), (111)

6: (1(12)), (2(11)), (112)

7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)

8: (1(23)), (2(13)), (3(12)), (123)

9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

gro[m_]:=If[Length[m]==1, m, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]];

Table[Length[gro[Flatten[MapIndexed[Table[#2, {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]]], {n, 20}]

CROSSREFS

Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653, A316654, A316656.

Sequence in context: A258861 A171038 A023395 * A318848 A193798 A101409

Adjacent sequences: A316652 A316653 A316654 * A316656 A316657 A316658

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 09 2018

EXTENSIONS

a(37)-a(40) from Robert Price, Sep 13 2018

STATUS

approved